4.4 The mean value theorem  (Page 3/7)

One application that helps illustrate the Mean Value Theorem involves velocity. For example, suppose we drive a car for 1 h down a straight road with an average velocity of 45 mph. Let $s(t)$ and $v\left(t\right)$ denote the position and velocity of the car, respectively, for $0\le t\le 1$ h. Assuming that the position function $s\left(t\right)$ is differentiable, we can apply the Mean Value Theorem to conclude that, at some time $c\in \left(0,1\right),$ the speed of the car was exactly

Corollaries of the mean value theorem

Let’s now look at three corollaries of the Mean Value Theorem. These results have important consequences, which we use in upcoming sections.

At this point, we know the derivative of any constant function is zero. The Mean Value Theorem allows us to conclude that the converse is also true. In particular, if $f^{\prime }\left(x\right)=0$ for all $x$ in some interval $I,$ then $f\left(x\right)$ is constant over that interval. This result may seem intuitively obvious, but it has important implications that are not obvious, and we discuss them shortly.

Proof

Since $f$ is differentiable over $I,$ $f$ must be continuous over $I.$ Suppose $f\left(x\right)$ is not constant for all $x$ in $I.$ Then there exist $a,b\in I,$ where $a\ne b$ and $f\left(a\right)\ne f\left(b\right).$ Choose the notation so that $a<b.$ Therefore,